Root datum

In mathematical group theory, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism. They were introduced by Michel Demazure in SGA III, published in 1970.

Definition

A root datum consists of a quadruple

$(X^{\ast },\mathrm{\Phi },X_{\ast },{\mathrm{\Phi }}^{\vee })$,

where

• $X^{\ast }$ and $X_{\ast }$ are free abelian groups of finite rank together with a perfect pairing between them with values in $\mathbb{\mathbb{Z}}$ which we denote by ( , ) (in other words, each is identified with the dual of the other).
• $\mathrm{\Phi }$ is a finite subset of $X^{\ast }$ and ${\mathrm{\Phi }}^{\vee }$ is a finite subset of $X_{\ast }$ and there is a bijection from $\mathrm{\Phi }$ onto ${\mathrm{\Phi }}^{\vee }$, denoted by $\alpha \mapsto {\alpha }^{\vee }$.
• For each $\alpha$, $(\alpha ,{\alpha }^{\vee })=2$.
• For each $\alpha$, the map $x\mapsto x-(x,{\alpha }^{\vee })\alpha$ induces an automorphism of the root datum (in other words it maps $\mathrm{\Phi }$ to $\mathrm{\Phi }$ and the induced action on $X_{\ast }$ maps ${\mathrm{\Phi }}^{\vee }$ to ${\mathrm{\Phi }}^{\vee }$)

The elements of $\mathrm{\Phi }$ are called the roots of the root datum, and the elements of ${\mathrm{\Phi }}^{\vee }$ are called the coroots. If $\mathrm{\Phi }$ does not contain $2\alpha$ for any $\alpha \in \mathrm{\Phi }$, then the root datum is called reduced.

The root datum of an algebraic group

If $G$ is a reductive algebraic group over an algebraically closed field $K$ with a split maximal torus $T$ then its root datum is a quadruple

$(X^{*},\mathrm{\Phi },X_{*},{\mathrm{\Phi }}^{\vee })$,

where

• $X^{*}$ is the lattice of characters of the maximal torus,
• $X_{*}$ is the dual lattice (given by the 1-parameter subgroups),
• $\mathrm{\Phi }$ is a set of roots,
• ${\mathrm{\Phi }}^{\vee }$ is the corresponding set of coroots.

A connected split reductive algebraic group over $K$ is uniquely determined (up to isomorphism) by its root datum, which is always reduced. Conversely for any root datum there is a reductive algebraic group. A root datum contains slightly more information than the Dynkin diagram, because it also determines the center of the group. For any root datum $(X^{*},\mathrm{\Phi },X_{*},{\mathrm{\Phi }}^{\vee })$, we can define a dual root datum $(X_{*},{\mathrm{\Phi }}^{\vee },X^{*},\mathrm{\Phi })$ by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots. If $G$ is a connected reductive algebraic group over the algebraically closed field $K$, then its Langlands dual group ${}^{L}G$ is the complex connected reductive group whose root datum is dual to that of $G$.

References

• Michel Demazure, Exp. XXI in SGA 3 vol 3
• T. A. Springer, Reductive groups, in Automorphic forms, representations, and L-functions vol 1 ISBN 0-8218-3347-2